scholarly journals On the longest runs in Markov chains

2018 ◽  
Vol 38 (2) ◽  
pp. 407-428 ◽  
Author(s):  
Zhenxia Liu ◽  
Xiangfeng Yang
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Generating Function ◽  
Cumulative Distribution Function ◽  
Moment Generating Function ◽  
Cumulative Distribution ◽  
Global Estimate ◽  
Success Run ◽  
Longest Success Run ◽  
The Moment

In the first n steps of a two-state success and failure Markov chain, the longest success run Ln has been attracting considerable attention due to its various applications. In this paper, we study Ln in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our method on the moment generating function is based on a global estimate of the cumulative distribution function of Ln proposed in this paper, and the proofs of the large deviations include the Gärtner–Ellis theorem and the moment generating function.

scholarly journals Deductibles and the Inverse Gaussian Distribution

1994 ◽  
Vol 24 (2) ◽  
pp. 319-323 ◽  
Author(s):  
Peter ter Berg
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Gaussian Distribution ◽  
Life Insurance ◽  
Beta Function ◽  
Moment Generating Function ◽  
Inverse Gaussian Distribution ◽  
Cumulative Distribution ◽  
Inverse Gaussian ◽  
The Moment

The calculation of mean claim sizes, in the presence of a deductible, is usually achieved through numerical integration. In case of a Lognormal or Gamma distribution, the quantities of interest can easily be expressed as functions of the cumulative distribution function, with modified parameters. This also applies to the F-distribution, where the incomplete Beta function enters the scene; see for instance the appendix in Hogg and Klugman (1984).The purpose of this paper is to derive an explicit formula for the first two moments of the Inverse Gaussian distribution, in the presence of censoring. For reasons of completeness we also consider truncation of the Inverse Gaussian distribution by an upper limit.The tractability of the derivation depends in a crucial way on two properties of the Inverse Gaussian distribution. Firstly, the cumulative distribution function of the Inverse Gaussian can be written as a simple function using the Normal probability integral. Secondly, the moment generating function of a censorized Inverse Gaussian distribution boils down to an expression containing the cumulative Inverse Gaussian distribution. This manifests itself most clearly in case of life insurance where the quantity of interest is the expectation of a present value. In case of non-life insurance, where the dimension of the Inverse Gaussian random variable is money instead of time, a further step is required: differentiation of the moment generating function.So, a natural order of this paper is to address ourselves first to the derivation for the life case and afterwards tackling the more laborious derivation for the non-life case.

Large deviations in the supercritical branching process

1993 ◽  
Vol 25 (04) ◽  
pp. 757-772 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Population Size ◽  
Branching Process ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Large Deviation Theory ◽  
Tail Behaviour ◽  
The Moment ◽  
Deviation Theory

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

Dual-Hop Fixed Gain Relaying Transmissions with Semi-Blind in Asymmetric Multipath/Shadowing Fading Channels

2015 ◽  
Vol 9 (1) ◽  
pp. 82-90
Author(s):  
Weijun Cheng ◽  
Teng Chen
Keyword(s):  
Fading Channels ◽  
Cumulative Distribution Function ◽  
Signal To Noise Ratio ◽  
Ergodic Capacity ◽  
Moment Generating Function ◽  
Symbol Error Rate ◽  
Cumulative Distribution ◽  
Simulation Results ◽  
End To End ◽  
The Moment

In this paper, we investigate the end-to-end performance of a dual-hop fixed gain relaying system with semiblind relay under asymmetric fading environments. In such environments, the wireless links of the considered system undergo asymmetric multipath/shadowing fading conditions, where one link is subject to only the Nakagami-m fading, the other link is subject to the composite Nakagami-lognormal fading which is approximated by using mixture gamma fading model. First, the cumulative distribution function (CDF), the moment generating function (MGF) and the moments of the end-to-end signal-to-noise ratio (SNR) are derived under two asymmetric scenarios. Then, novel closed-form expressions of the outage probability, the average end-to-end SNR, the symbol error rate and the ergodic capacity for the dual-hop system are obtained based on the CDF and the MGF, respectively. Finally, some numerical and simulation results are shown and discussed to validate the accuracy of the analytical results under different scenarios, such as varying average SNR, fading parameters per hop, the choice of the semi-blind gain and the location of relaying nodes.

scholarly journals Effects of the Age Process on Aggregate Discounted Claims

Risks ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 106
Author(s):  
Ghislain Léveillé ◽  
Ilie-Radu Mitric ◽  
Victor Côté
Keyword(s):  
Distribution Function ◽  
Generating Function ◽  
Moment Generating Function ◽  
Past Experience ◽  
Joint Moments ◽  
The Past ◽  
Insurance Portfolio ◽  
The Moment ◽  
Age Process

In this document, we examine the effects of the age process on aggregate discounted claims by studying the conditional raw and joint moments, the moment generating function and the distribution function of the increments of compound renewal sums with discounted claims, taking into account the past experience of an insurance portfolio.

Large deviations in the supercritical branching process

1993 ◽  
Vol 25 (4) ◽  
pp. 757-772 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Population Size ◽  
Branching Process ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Large Deviation Theory ◽  
Tail Behaviour ◽  
The Moment ◽  
Deviation Theory

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

scholarly journals Inverse Analogue of Ailamujia Distribution with Statistical Properties and Applications

2020 ◽  
pp. 36-46
Author(s):  
Ahmad Aijaz ◽  
Afaq Ahmad ◽  
Rajnee Tripathi
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Real Life ◽  
Likelihood Estimation ◽  
Moment Generating Function ◽  
Statistical Properties ◽  
Cumulative Distribution ◽  
Life Data ◽  
Real Life Data ◽  
Method Of Maximum Likelihood

The present paper deals with the inverse analogue of Ailamujia distribution (IAD). Several statistical properties of the newly developed distribution has been discussed such as moments, moment generating function, survival measures, order statistics, shanon entropy, mode and median .The behavior of probability density function (p.d.f) and cumulative distribution function (c.d.f) are illustrated through graphs. The parameter of the newly developed distribution has been estimated by the well known method of maximum likelihood estimation. The importance of the established distribution has been shown through two real life data.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

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